Let $A$ be a Square Matrix where $A \in M^C_{n \times n}$ and $P(x)$ be its characteristic polynomial.
$A$ is also nilpotent.
Is it true if $A$ is a matrix $A_{n, n} $ in the complex matrices field, which is nilpotent, then its characteristic polynomial has a degree $n^2$?
If it's not true, when does it happen? What are the conditions to such case?
Thanks,
Alan
Best Answer
The characteristic polynomial of a matrix is $\mathrm{det}(A-\lambda I)$, so this determinant will have one factor of $\lambda$ for each entry in the diagonal of $A - \lambda I$, so the degree of the characteristic polynomial of an $n\times n$ matrix $A$ is $n$ (corresponding to one multiple of $\lambda$ for each row (or column)).